We draw the reader's attention to the Carleman contraction principle, see [1]}, [2]}, [3]}, [4]} to solve related problems to Problem REF .
Like the proposed Carleman-Newton method, the Carleman contraction principle is very powerful since it can quickly solve nonlinear inverse problems without requesting a good initial guess.
The new point of our paper in comparison to the cited publications above is that we allow \(F\)  to depend on \(\nabla u\)  and we study the case when data contains noise.
The case when \(F\)  depends on \(\nabla u\)  arises from the field of Hamilton-Jacobi equations of the form \(u_t = \epsilon \Delta u + F({\bf x}, u, \nabla u)\) , \(0 < \epsilon \ll 1\)  where the term \(\epsilon \Delta u\)  serves as the viscosity term. Hamilton-Jacobi equations are important in many fields; e.g. game theory and light propagation.
Therefore, this gradient dependent case is worth studying.
Rather than using the Carleman contraction principle as in [1]}, [2]}, [3]}, [4]}, we develop the Carleman-Newton method to solve the inverse problem under consideration.
The Carleman-Newton method in this paper is stronger than the one introduced in [9]}.
In fact, we can prove a Lipschitz stability result of our proposed method with respect to the \(H^2\)  norm while the convergence in the cited papers above, as well as [9]}, is with respect to the \(H^1\)  norm.
Another new point of this paper is that in the current paper, we study the noise analysis, which was missing in [9]}.
